The Scriptural Teaching of the Trinity

There is but one God -
 
Deut. 6:4Hear, O Israel: The Lord our God is One.
Deut. 32:39Now see that I, even I, am He, and there is no God besides Me; I kill and I make alive; I wound and I heal; nor is there any who can deliver from My hand.
1 Tim. 2:5For there is one God and one Mediator between God and men, the Man Christ Jesus.
1 Cor. 8:4There is no other God but one.

There is plurality within this one God -

As Found in the Old Testament
elohimFound over 2,500 times in the Old Testament
Gen. 1:26Let Us make man in our image
Gen. 3:22Behold, the man has become like one of Us, to know good and evil
Gen. 11:7Come, let Us go down and there confuse their language
Isa. 48:16And now the Lord God and His Spirit have sent Me
Isa. 61:1The Spirit of the Lord God is upon Me
Isa. 63:9-10In all their affliction he was afflicted, and the angel of his presence saved them: in his love and in his pity he redeemed them; and he bare them, and carried them all the days of old.  But they rebelled, and vexed his holy Spirit: therefore he was turned to be their enemy, [and] he fought against them.
Creation and Providence Ascribed to the Father, Word, and Spirit in the Old Testament
 
Father
Gen. 1:1In the beginning God created the heaven and the earth.
Job 34:12-15Yea, surely God will not do wickedly, neither will the Almighty judgment.  Who hath given him a charge over the earth? or who hath disposed the whole world?  If he set his heart upon man, [if] he gather unto himself his spirit and his breath; All flesh shall perish together, and man shall turn again unto dust.
Word
Gen. 1:3And God said, Let there be light: and there was light.
Psalm 33:6, 9By the word of the LORD were the heavens made; and all the host of them by the breath of his mouth. . . . For he spake, and it was [done]; he commanded, and it stood fast
Spirit
Gen. 1:2And the earth was without form, and void; and darkness [was] upon the face of the deep. And the Spirit of God moved upon the face of the waters.
Psalm 104:30Thou sendest forth thy spirit, they are created: and thou renewest the face of the earth.
As Found in the New Testament
 
Luke 1:35And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.
Matt. 3:16-17And Jesus, when he was baptized, went up straightway out of the water: and, lo, the heavens were opened unto him, and he saw the Spirit of God descending like a dove, and lighting upon him:  And lo a voice from heaven, saying, This is my beloved Son, in whom I am well pleased.
1 Cor. 12:4-6The same Spirit . . . the same Lord . . . the same God
2 Cor. 13:14The grace of the Lord Jesus, the love of God, and the fellowship of the Holy Spirit be with you all
1 Pet. 1:2Elect according to the foreknowledge of God the Father, in sanctification of the Spirit, for obedience and sprinkling of the blood of Jesus Christ
Matt. 28:19Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost
Creation and Providence Ascribed to the Father, Son, and Holy Spirit in the New Testament
 
Father
Matt. 7:11If ye then, being evil, know how to give good gifts unto your children, how much more shall your Father which is in heaven give good things to them that ask him?
Acts 17:28For in him we live, and move, and have our being; as certain also of your own poets have said, For we are also his offspring.
Son
John 1:3All things were made by him; and without him was not any thing made  that was made.
Col. 1:15-17Who is the image of the invisible God, the firstborn of every creature: For by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether [they be] thrones, or dominions, or principalities, or powers: all things were created by him, and for him: And he is before all things, and by him all things consist.
Holy Spirit
Matt. 1:18; 4:1Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost. . . . Then was Jesus led up of the Spirit into the wilderness to be tempted of the devil.
Rm. 1:4And declared [to be] the Son of God with power, according to the spirit of holiness, by the resurrection from the dead

Two major heresies in the history of the Church -

The orthodox teaching of the Trinity -
 

The Apostles Creed -

The Nicene Creed

The Athanasian Creed